A) \[\left( -\frac{\pi }{2},0 \right)\]
B) \[(0,\pi )\]
C) \[\left( \pi ,\frac{3\pi }{2} \right)\]
D) \[\left( 0,\frac{\pi }{2} \right)\]
Correct Answer: B
Solution :
Let f(x) = sin x + x cos x Consider, \[g(x)=\int\limits_{0}^{x}{(\sin t+t\cos )}dt=t\sin t]_{0}^{x}=x\sin x\] g (x) = x sin x which is differentiable Now g (0) = 0 and g \[(\pi )=0\], using Rolles Theorem \[\exists \] at least one \[c\in (0,\pi )\] such that \[g'(c)=0\] i.e. c cos c + sin c = 0 for at least one \[c\in (0,\pi )\]
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